417 



must be a circle. From these equations all the usual theory of the 

 straight line and circle may be readily deduced, and all ambiguity 

 respecting the representation of direction by the signs (-}-) and ( ) 

 may be removed. Thus if the loci of 



OP=# . Ol+y . OJ, y=ax+b, 

 and OF=^ 



intersect in the point Q, and OQ X be the unit radius on the J side 

 of OI (produced both ways), determined by the equation 



then V(l+ 2 )-OQ will =bOQ l . And a line drawn from the 

 locus of P parallel to OQ to pass through the point X, where 

 OX=;T 1 .OI+ y r OJ, will be represented in magnitude and direc- 



tion by 



y^-a^-b 



vo+o* 



From this result the usual theory of anharmonic ratios is immediately 

 deducible without any fresh * convention ' respecting the signs ( + ) 

 and ( ). 



As every locus has two equations, each equation requires separate 

 consideration. The investigations concerning the abstract equation 

 remain nearly the same as in the usual theory. When the abstract 

 equation is given indirectly by the elimination of two constants 

 between three equations, the result corresponds to the locus of the 

 intersection of two curves varying according to a known law, 

 ' coordinates proper,' leading in its simplest forms, first, to Descartes' 

 original conception of curves generated by the intersection of straight 

 lines moving according to a given law parallel to two given straight 

 lines, and secondly, to Pliicker's 'point coordinates.' The true 

 relation of Pliicker's 'line coordinates' to the ordinary system is 

 immediately apparent on comparing the two sets of equations : 



(1) Concrete OP=#. Ol-j-y . OJ 



Abstract F(a?,y) = 0, 



( 2 ) First abstract y~ax+b 



Second abstract F(a, &)=0. 



The second set of equations determines a curve by supposing a and b 

 to vary, then eliminating a and b } and referring the ultimate abstract 



.2 G2 



